To solve the problem step by step, we need to determine how much each person (A, B, C, and D) paid for the gift worth Rs 60, given the conditions about their contributions.
### Step 1: Define the contributions
Let:
- A's contribution = A
- B's contribution = B
- C's contribution = C
- D's contribution = D
According to the problem:
- A pays \( \frac{1}{2} \) of what B, C, and D pay.
- B pays \( \frac{1}{3} \) of what A, C, and D pay.
- C pays \( \frac{1}{4} \) of what A, B, and D pay.
### Step 2: Set up equations based on the contributions
From the information given, we can set up the following equations:
1. \( A = \frac{1}{2}(B + C + D) \)
2. \( B = \frac{1}{3}(A + C + D) \)
3. \( C = \frac{1}{4}(A + B + D) \)
### Step 3: Express B, C, and D in terms of A
From equation (1):
\[ 2A = B + C + D \]
So,
\[ B + C + D = 2A \] (Equation 4)
From equation (2):
\[ 3B = A + C + D \]
So,
\[ A + C + D = 3B \] (Equation 5)
From equation (3):
\[ 4C = A + B + D \]
So,
\[ A + B + D = 4C \] (Equation 6)
### Step 4: Substitute equations
Now we have three equations (4, 5, and 6). We can express B and C in terms of A using these equations.
From Equation 4, we can express D:
\[ D = 2A - B - C \]
Substituting D in Equation 5:
\[ A + C + (2A - B - C) = 3B \]
This simplifies to:
\[ 3A - B = 3B \]
Thus:
\[ 3A = 4B \]
So,
\[ B = \frac{3}{4}A \] (Equation 7)
Now substituting B in Equation 6:
\[ A + \frac{3}{4}A + D = 4C \]
This simplifies to:
\[ \frac{7}{4}A + D = 4C \]
So,
\[ D = 4C - \frac{7}{4}A \] (Equation 8)
### Step 5: Use total contribution
Now we know that:
\[ A + B + C + D = 60 \]
Substituting B and D from Equations 7 and 8:
\[ A + \frac{3}{4}A + C + (4C - \frac{7}{4}A) = 60 \]
This simplifies to:
\[ A + \frac{3}{4}A - \frac{7}{4}A + 5C = 60 \]
Combining terms:
\[ -\frac{3}{4}A + 5C = 60 \]
Multiplying through by 4 to eliminate the fraction:
\[ -3A + 20C = 240 \] (Equation 9)
### Step 6: Solve for C
Now we can express C in terms of A:
\[ 20C = 3A + 240 \]
So,
\[ C = \frac{3A + 240}{20} \]
### Step 7: Substitute C back to find A
Substituting C back into Equation 4:
\[ B + C + D = 2A \]
Using B from Equation 7 and D from Equation 8, we can find A.
After substituting and simplifying, we find:
- A = Rs 20
- B = Rs 15
- C = Rs 12
- D = Rs 13
### Final Answer
Thus, the amount paid by D is **Rs 13**.
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